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Understanding the Square of a Binomial: (a + b)^2 = a^2 + b^2 + 2ab
In algebra, a binomial is an expression with two terms, often involving variables. The square of a binomial, represented as (a + b)², is a common algebraic expression that arises in various mathematical applications.
The Expansion of (a + b)²
The expression (a + b)² represents the product of (a + b) with itself:
(a + b)² = (a + b)(a + b)
To expand this expression, we use the distributive property of multiplication:
- Multiply the first term of the first binomial (a) with each term in the second binomial:
- a * a = a²
- a * b = ab
- Multiply the second term of the first binomial (b) with each term in the second binomial:
- b * a = ab
- b * b = b²
Adding all the terms together, we get:
(a + b)² = a² + ab + ab + b²
Combining like terms, we obtain the final expansion:
(a + b)² = a² + 2ab + b²
Understanding the Formula
This formula reveals a pattern:
- The first term: a² is the square of the first term (a) of the binomial.
- The second term: 2ab is twice the product of the first term (a) and the second term (b) of the binomial.
- The third term: b² is the square of the second term (b) of the binomial.
Applications of the Formula
The formula (a + b)² = a² + 2ab + b² has numerous applications in algebra, geometry, and other areas of mathematics, including:
- Simplifying algebraic expressions: The formula can be used to simplify expressions involving the square of a binomial.
- Factoring expressions: The formula can be used to factor quadratic expressions.
- Solving equations: The formula can be used to solve equations involving the square of a binomial.
- Deriving other formulas: The formula can be used to derive other algebraic identities, such as the difference of squares formula.
Example
Let's consider an example:
(x + 3)²
Applying the formula:
(x + 3)² = x² + 2(x)(3) + 3²
Simplifying:
(x + 3)² = x² + 6x + 9
Therefore, the expansion of (x + 3)² is x² + 6x + 9.
Conclusion
The formula (a + b)² = a² + b² + 2ab is a fundamental algebraic identity that is widely used in various mathematical contexts. Understanding the expansion and application of this formula is essential for mastering algebraic concepts and solving problems involving binomials.